From unidirectional to grid: Generation and precise control of multi-scroll chaotic attractors

Artificial neural networks provide crucial model support for understanding the complex functions of the brain. As a typical recurrent structure, the coupling of Hopfield neural networks (HNN) with memristors becomes a research hotspot. This paper proposes a memristor incorporating a multi-piecewise arctangent function, coupled with a three-dimensional HNN to construct a memristive HNN model. First, a single arctangent function-based memristor-HNN(SAFM-HNN) is constructed to achieve unidirectional replication of multi-scroll attractors. Subsequently, a second memristor is introduced to build a double arctangent function-based memristor-HNN(DAFM-HNN), enabling bidirectional replication and forming grid multi-scroll attractors. Since the DAFM-HNN inherits many dynamical properties from the SAFM-HNN, the DAFM-HNN is taken as an example in this paper. Its equilibrium points are analyzed, and parameter effects are examined through bifurcation diagrams (BD), Lyapunov exponents (LEs), Spectral Entropy (SE), and discharge plots. The system exhibits initial-value-dependent offset-boosting and controllable amplitude characteristics, supporting the coexistence of multiple multi-scroll attractors. Finally, its digital circuit feasibility is validated via a digital signal processor (DSP) platform. These results offer new insights into brain function exploration and bionics.